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Factor. \( 9 v^{10}-6 v^{5}+1 \) \( 9 v^{10}-6 v^{5}+1=\square \)

Ask by Cox May. in the United States
Feb 24,2025

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Answer

\( 9v^{10} - 6v^{5} + 1 = (3v^{5} - 1)^2 \)

Solution

Let u = v⁵. Then the expression becomes:   9u² – 6u + 1 This is a perfect square trinomial since:   (3u – 1)² = 9u² – 6u + 1 Returning to v, we have:   (3v⁵ – 1)² Thus,   9v¹⁰ – 6v⁵ + 1 = (3v⁵ – 1)².

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The Deep Dive

To factor \( 9 v^{10}-6 v^{5}+1 \), we can view it as a quadratic in terms of \( v^5 \). Let \( x = v^5 \); then our expression becomes \( 9x^2 - 6x + 1 \). This can be factored as: \[ (3x - 1)^2 \] Substituting back \( x = v^5 \), we have: \[ 9 v^{10} - 6 v^{5} + 1 = (3v^5 - 1)^2 \] Who knew? Quadratics can grow up to be perfect squares too! In real-world applications, recognizing patterns in algebra can make solving complex equations much simpler. For instance, architects often need to work with polynomial equations when designing structures. Understanding how to factor efficiently can save time and aid in constructing accurate models of physical spaces!

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